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The Robertson–Seymour theorem states that finite undirected graphs and graph minors form a well-quasi-ordering. The graph minor relationship does not contain any infinite descending chain, because each contraction or deletion reduces the number of edges and vertices of the graph (a non-negative integer). [8]
An edge contraction is an operation that removes an edge from a graph while simultaneously merging the two vertices it used to connect. An undirected graph H is a minor of another undirected graph G if a graph isomorphic to H can be obtained from G by contracting some edges, deleting some edges, and deleting some isolated vertices.
By Robertson–Seymour theorem, any set of finite graphs contains only a finite number of minor-minimal elements. In particular, the set of "yes" instances has a finite number of minor-minimal elements. Given an input graph G, the following "algorithm" solves the above problem: For every minor-minimal element H: If H is a minor of G then return ...
A minor of a graph G is any graph H that is isomorphic to a graph that can be obtained from a subgraph of G by contracting some edges. If G does not have a graph H as a minor, then we say that G is H-free. Let H be a fixed graph. Intuitively, if G is a huge H-free graph, then there ought to be a "good reason" for this.
In mathematics, a simple subcubic graph (SSCG) is a finite simple graph in which each vertex has a degree of at most three. Suppose we have a sequence of simple subcubic graphs G 1, G 2, ... such that each graph G i has at most i + k vertices (for some integer k) and for no i < j is G i homeomorphically embeddable into (i.e. is a graph minor of) G j.
Since every minor of a planar graph is itself planar, this gives a planar cover of the minor G. Because the graphs with planar covers are closed under the operation of taking minors, it follows from the Robertson–Seymour theorem that they may be characterized by a finite set of forbidden minors. [7] A graph is a forbidden minor for this ...
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